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Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Bertram Düring, Michel Fournié, Ansgar Jüngel (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Bertram Düring, Michel Fournié, Ansgar Jüngel (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation

Guillaume Legendre, Takéo Takahashi (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem

Florian Mehats (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...

Convergence of a numerical scheme for a nonlinear oblique derivative boundary value problem

Florian Mehats (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present here a discretization of a nonlinear oblique derivative boundary value problem for the heat equation in dimension two. This finite difference scheme takes advantages of the structure of the boundary condition, which can be reinterpreted as a Burgers equation in the space variables. This enables to obtain an energy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of this problem and a numerical study of the stability of the scheme, which appears...

Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation

Daniel Matthes, Horst Osberger (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

Abdur Rashid, Shakaib Akram (2010)

Applications of Mathematics

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Sébastien Martin, Julien Vovelle (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial...

Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Nicolas Besse, Dietmar Kröner (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition Δ t ∼ h 4 / 3 , we obtain error estimates in L 2 of order 𝒪 ( Δ t 2 + h m + 1 / 2 ) where m is the degree of the local polynomials.

Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system

Nicolas Besse, Dietmar Kröner (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition Δ t ∼ h 4 / 3 , we obtain error estimates in L2 of order 𝒪 ( Δ t 2 + h m + 1 / 2 ) where m is the degree of the local polynomials.

Convergence of Rothe's method in Hölder spaces

Norio Kikuchi, Jozef Kačur (2003)

Applications of Mathematics

The convergence of Rothe’s method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe’s approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.

Convergence of the finite element method applied to an anisotropic phase-field model

Erik Burman, Daniel Kessler, Jacques Rappaz (2004)

Annales mathématiques Blaise Pascal

We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H 1 -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not...

Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Béla J. Szekeres, Ferenc Izsák (2017)

Applications of Mathematics

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on ℝ 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated...

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